Image encryption and visual hiding method based on chaotic neural network and compressed sensing

Abstract

The application belongs to the technical field of image encryption, and discloses an image encryption and visual hiding method based on chaotic neural network and compressed sensing, which comprises the following steps: constructing a fractional order heterogeneous chaotic neural network, and making the encryption system in a complex chaotic state based on the fractional order chaotic neural network; taking an initial value as a starting point, generating a chaotic sequence required for encryption through numerical iteration; based on the chaotic sequence, carrying out two-dimensional compression and hybrid encryption on a plaintext image, and embedding the plaintext image into a color carrier to obtain a ciphertext image; extracting the ciphertext from the ciphertext image, and carrying out reverse decryption and high-quality reconstruction based on a 2DPG-ED algorithm to restore a two-dimensional plaintext image. The application not only realizes deep confusion and dimension reduction transmission of the ciphertext, but also gives the ciphertext a normal visual form, and comprehensively improves the security, concealment and attack resistance robustness of the encryption system.

Description

Technical Field

[0001] This invention belongs to the field of image encryption technology, specifically relating to image encryption and visual hiding methods based on chaotic neural networks and compressed sensing. Background Technology

[0002] With the development of information technology, the issues of secure transmission and transmission rate limitations of image data have become increasingly prominent. Although existing image encryption schemes based on chaotic systems and compressed sensing can achieve data compression and protection to a certain extent, they still cannot meet the actual high-security engineering requirements in terms of robustness and visual concealment in dealing with encrypted transmission.

[0003] Specifically, the shortcomings of the existing technology are: (1) The randomness of the integer order chaotic system is simple, the key space is limited, and it is easy to crack by statistical analysis; (2) The encryption mechanism is simple (such as diffusion and scrambling based on chaos, DNA encoding) and it is difficult to achieve deep encryption. The pixel correlation is high and the storage pressure is high; (3) The ciphertext output by the existing encryption scheme is mostly in the form of noise snowflakes. It is easy to expose confidential data during transmission, attract malicious interception and targeted decryption attacks, and lacks visual concealment.

[0004] Therefore, there is an urgent need for a triple encryption scheme that combines highly complex chaotic systems with compression, encryption, and hiding to improve the security and robustness of image encryption. Summary of the Invention

[0005] This invention aims to address the shortcomings of existing technologies by proposing an image encryption and visual hiding method based on chaotic neural networks and compressed sensing. It utilizes a chaotic system to generate chaotic sequences; employs 2D compressed sensing (2D CS) to effectively reduce the transmission and storage pressure of ciphertext; combines double-layer Josephus scrambling (DLJPS) with adjacent diagonal diffusion algorithm to achieve pixel-level obfuscation and enhance anti-attack capabilities; uses 2D integer wavelet transform (2D IWT) to generate low-frequency (LL) and high-frequency (LH, HL, HH) subbands, embedding the ciphertext sequentially into the frequency domain coefficients of the red, green, and blue channels of the color carrier in a least significant bit (LSB) manner, thus giving it a normal visual appearance to evade malicious interception; and utilizes the 2D projection gradient embedding decryption (2DPG-ED) algorithm for high-quality, noise-resistant reconstruction.

[0006] To achieve the above objectives, the present invention provides the following solution:

[0007] An image encryption and visual hiding method based on chaotic neural networks and compressed sensing includes the following steps:

[0008] Construct a fractional-order heterogeneous chaotic neural network, and use the fractional-order chaotic neural network to put the encryption system into a complex chaotic state;

[0009] Starting with an initial value, a chaotic sequence required for encryption is generated through numerical iteration;

[0010] Based on the chaotic sequence, the plaintext image is subjected to two-dimensional compression and hybrid encryption, and then embedded into a color carrier to obtain a encrypted image;

[0011] Ciphertext is extracted from the ciphertext image, and reverse decryption and high-quality reconstruction are performed based on the 2DPG-ED algorithm to restore the two-dimensional plaintext image.

[0012] Preferably, the method for constructing the fractional-order heterogeneous chaotic neural network includes:

[0013] Define the fractional Caputo operator:

[0014] ,

[0015] Where q represents the fractional order. This represents the gamma function, where m represents the smallest integer greater than or equal to q. Describe the objective function The lower limit of time is The global operator of q-order Caputo fractional calculus at time t. The lower limit of time is indicated as Riemann-Liouville fractional integral operator of order mq This represents a standard m-order integer differential operator, where t represents the current evolution time. This represents the m-th derivative of the objective function with respect to the historical time variable τ. This represents the denominator of the power-law memory kernel function in fractional calculus formulas. This means that when the order q is exactly equal to an integer m, Caputo fractional calculus seamlessly degenerates into the objective function. The standard m-th order ordinary differential with respect to time t;

[0016] A sinusoidal magnetically controlled memristor model is introduced to simulate the synaptic crossing connections of neurons in the brain region. The sinusoidal magnetically controlled memristor model is as follows:

[0017] ,

[0018] Where i represents the output current, v represents the input voltage, φ represents the internal magnetic flux state variable, α represents the activation gradient of the memristor, and k represents the internal control parameters of the memristor. This indicates that the state variable φ is from The initial q-order fractional derivative;

[0019] By combining different nonlinear activation functions, and based on the fractional-order Caputo operator and the sinusoidal magnetically controlled memristor model, the fractional-order heterogeneous chaotic neural network is constructed:

[0020] ,

[0021] in, and This represents the neuron state variables using the Hardtanh activation function. and Let represent the neuron state variables using the LeakyReLU activation function, u represent the internal state variables of the memristor, p represent the coupling strength of the memristor, I represent the external input bias current, α represent the memristor activation gradient, and k represent the internal parameters of the memristor. This indicates that the state variable x starts from the initial time. The q-order Caputo fractional differential operator.

[0022] Preferably, the Hardtanh activation function is:

[0023] ;

[0024] The LeakyReLU activation function is:

[0025] ,

[0026] Where β represents the leakage coefficient.

[0027] Preferably, the method for generating the chaotic sequence includes:

[0028] Starting with given initial values ​​and system parameters, the fractional-order heterogeneous chaotic neural network is discretized and iteratively solved.

[0029] Discarding the transient processes of the first few iterations, extract the sequence of state variables of the system to obtain several sets of chaotic sequences of independent length.

[0030] Preferably, the method for obtaining the dense image includes:

[0031] The plaintext image is sorted by one-dimensional position index using the chaotic sequence to obtain a pre-set scrambled image;

[0032] A measurement matrix is ​​constructed based on the chaotic sequence. The measurement matrix is ​​optimized by SVD singular value decomposition. Then, the preset scrambled image is subjected to 2D CS and linear quantization to obtain the first compressed ciphertext matrix.

[0033] The step size and direction parameters are dynamically generated using the chaotic sequence, and a double-layer Josephus scrambling with variable step size and direction is performed on the first compressed ciphertext matrix.

[0034] The scrambled matrix is ​​subjected to the first round of diagonal diffusion from the top left to the bottom right. Then the matrix is ​​rotated 180 degrees to perform the second round of diagonal diffusion to enhance the encryption scheme’s resistance to differential attacks, and finally the first pure ciphertext matrix is ​​obtained.

[0035] Perform 2D IWT on the RGB three channels of the color carrier image to obtain the low-frequency subband and the high-frequency subband;

[0036] The least significant bit algorithm is used to convert the first pure ciphertext matrix into a bit stream and adaptively embed it into the high-frequency subband, while 2K correction is performed to compensate for amplitude error.

[0037] Perform inverse 2D IWT on each modified frequency domain subband to generate and output the color-coded dense image.

[0038] Preferably, the method for diagonal diffusion is as follows:

[0039] ,

[0040] in, This represents an image before a single round of diffusion. Let represent the encrypted image obtained after one round of diffusion, K represent the diffusion mask matrix generated by the chaotic sequence, M represent the size of the image, and (m, n) represent the coordinates of the current pixel.

[0041] Preferably, the method for restoring the two-dimensional plaintext image includes:

[0042] Perform 2D IWT on the received ciphertext image, extract the ciphertext bitstream from the high-frequency subband and reassemble it to obtain a second pure ciphertext matrix;

[0043] Based on the synchronously generated chaotic sequence, reverse diagonal diffusion and reverse DLJPS are sequentially performed on the extracted second pure ciphertext matrix to recover the second compressed ciphertext matrix containing quantization error.

[0044] Using the second compressed ciphertext matrix as the observation target, the 2DPG-ED algorithm is used to perform joint decompression and reconstruction. In the iterative gradient descent, the total variation model is combined to perform sparse projection in the plaintext domain, filter out reconstruction artifacts and channel noise, and output a high-quality decrypted image.

[0045] A reverse index sequence is generated based on the chaotic sequence, and the decrypted image is reverse sorted and scrambled to restore it, outputting the final two-dimensional plaintext image.

[0046] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0047] This invention constructs a highly complex fractional heterogeneous chaotic neural network, combining 2D CS, DLJPS and adjacent diagonal diffusion, as well as a frequency domain steganography mechanism based on 2D IWT. This not only achieves deep obfuscation and dimensionality reduction transmission of ciphertext, but also endows the ciphertext with a normal visual form, comprehensively improving the security, concealment and anti-attack robustness of the encryption system. The pure encrypted image has an information entropy of 7.9969 (extremely close to the theoretical maximum value of 8 for an 8-bit image), with highly uniform and random pixel distribution, effectively resisting differential and statistical attacks. The NPCR (pixel change rate) is 99.6307% and the UACI (change intensity) is 33.4357%, close to the theoretical ideal value, highly sensitive to plaintext single-pixel modifications, and exhibiting a perfect avalanche effect. Faced with 80×80 area data cropping attacks or salt-and-pepper noise attacks with an intensity of 0.01, the decrypted and reconstructed image based on the 2DPG-ED algorithm can still clearly retain the main outline and visual information of the original image. The average PSNR of the encrypted image and the original color carrier image is as high as 52.84dB, and the MSSIM is 0.9974, achieving near-perfect visual invisibility. At the same time, under the condition of a compression rate of 0.25, the PSNR of the decrypted and reconstructed image can still reach 36.2414dB, achieving extremely high-quality noise-resistant decompression and original information restoration. Attached Figure Description

[0048] To more clearly illustrate the technical solution of the present invention, the drawings used in the embodiments are briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0049] Figure 1 This is a schematic diagram of the method flow according to an embodiment of the present invention;

[0050] Figure 2 This is a q-bifurcation diagram according to an embodiment of the present invention;

[0051] Figure 3 Embodiments of the present invention Schematic diagram of a two-parameter suction basin;

[0052] Figure 4 Embodiments of the present invention Phase diagram

[0053] Figure 5 This is a schematic diagram of the encryption process according to an embodiment of the present invention;

[0054] Figure 6 This is a schematic diagram of the decryption process according to an embodiment of the present invention. Detailed Implementation

[0055] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all of them. All other embodiments obtained by a person of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0056] To make the above objectives, features and advantages of the present invention more obvious and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0057] Embodiment:

[0058] In this embodiment, as shown in Figure 1 , the image encryption and visual hiding method based on chaotic neural network and compressed sensing includes the following steps:

[0059] Step S1. Construct a fractional-order heterogeneous chaotic neural network, and make the encryption system in a complex chaotic state based on the fractional-order chaotic neural network.

[0060] In this embodiment, the constructed fractional-order neural network is a five-dimensional fractional-order heterogeneous Hopfield neural network (FHHHNN) coupled with a sinusoidal magnetron memristor. The method for constructing the fractional-order heterogeneous chaotic neural network includes:

[0061] There are many definitions of continuous-time fractional-order differential operators, and the Caputo operator definition is commonly used in engineering. Define the fractional-order Caputo operator:

[0062] ,

[0063] where q represents the fractional order; represents the Gamma function; m represents the smallest integer greater than or equal to q (that is, rounding up q, satisfying m-1<q<m or q=m); represents the q-order Caputo fractional calculus integral operator of the target function when the lower time limit is ; represents the Riemann-Liouville fractional integral operator with the lower time limit of and the order of m-q; represents the standard m-order integer-order differential operator (the whole means first calculating the m-order ordinary derivative of the target function, and then performing the m-q order fractional integration on it, that is, when the fractional order is greater than 1, the differentiation can be split into ordinary differentiation and fractional differentiation with an order greater than 0 and less than 1); t represents the independent argument of the system, that is, the current evolution time; represents the m-order derivative of the target function with respect to the historical time variable τ; ​​The term represents the denominator of the power-law memory kernel function in the fractional calculus formula. This term makes the evolution of the current state of the system influenced by all past historical states, reflecting the global memory characteristic of the fractional system. This means that when the order q is exactly equal to an integer m, Caputo fractional calculus seamlessly degenerates into the objective function. The standard m-th order ordinary differential with respect to time t.

[0064] Memristors are nonlinear devices with non-volatility and local activating properties, which can easily guide neural network systems into a chaotic state. A sinusoidal magnetically controlled memristor model is introduced to simulate the synaptic crossing connections of neurons in the brain. The sinusoidal magnetically controlled memristor model is as follows:

[0065] ,

[0066] Where i represents the output current, v represents the input voltage, φ represents the internal magnetic flux state variable, α represents the activation gradient of the memristor, and k represents the internal control parameters of the memristor. This indicates that the state variable φ is from The initial q-order fractional derivative;

[0067] Construct the activation functions Hardtanh(x) and LeakyReLU(x) for the dynamic equations of heterogeneous neurons:

[0068] ;

[0069] ,

[0070] Where β represents the slope coefficient of the negative half-axis of LeakyReLU, i.e., the leakage coefficient.

[0071] By combining different nonlinear activation functions, a fractional-order heterogeneous chaotic neural network is constructed based on the fractional-order Caputo operator and the sinusoidal magnetically controlled memristor model:

[0072] ,

[0073] in, and This represents the neuron state variables using the Hardtanh activation function. and Let represent the neuron state variables using the LeakyReLU activation function, u represent the internal state variables of the memristor, p represent the coupling strength of the memristor, I represent the external input bias current, α represent the memristor activation gradient, and k represent the internal parameters of the memristor. This indicates that the state variable x starts from the initial time. The q-order Caputo fractional differential operator.

[0074] In this embodiment, the network model parameters are set as follows: q=0.85, α=1, β=0.01, p=1.5, k=0.1, I=1. By analyzing the bifurcation diagram of the system as parameter q changes, combined with... Dual-parameter suction basin and Phase diagram, such as Figure 2 , Figure 3 , Figure 4 As shown, the verification system is in a complex chaotic state and has extreme initial value sensitivity and coexistence attractor properties.

[0075] Step S2. Starting from the initial value, generate the chaotic sequence required for encryption through numerical iteration.

[0076] Methods for generating chaotic sequences include: starting with given initial values ​​and system parameters, performing discretized iterative solutions on a fractional-order heterogeneous chaotic neural network; discarding the transient processes of the first few iterations, extracting the system's state variable sequence, and obtaining several sets of chaotic sequences of independent lengths.

[0077] In this embodiment, the workflow of the chaotic sequence generation module includes: starting from an initial state , , , , Starting from this point, we iterate together. The process is repeated 1000 times (where the plaintext image is N×N and the compressed ciphertext image is M×M, M=N / 2), and the first 1000 transient processes are discarded to eliminate transient effects; the retained state variable sequence is discretized and normalized to generate 5 independent chaotic pseudo-random sequences (chaotic sequence S1, chaotic sequence S2, chaotic sequence S3, chaotic sequence S4, chaotic sequence S5).

[0078] Step S3. Based on the chaotic sequence, perform two-dimensional compression and hybrid encryption on the plaintext image, and embed it into the color carrier to obtain the encrypted image.

[0079] The method for obtaining the encrypted image includes: sorting the plaintext image using a one-dimensional position index using a chaotic sequence to obtain a pre-scrambled image; constructing a measurement matrix based on the chaotic sequence, optimizing the measurement matrix using SVD singular value decomposition, and then performing 2D CS and linear quantization on the pre-scrambled image to obtain a first compressed ciphertext matrix; dynamically generating step size and direction parameters using a chaotic sequence, and performing variable step size and direction double-layer Josephus scrambling on the first compressed ciphertext matrix; performing a first round of diagonal diffusion from the top left to the bottom right on the scrambled matrix, followed by rotating the matrix 180 degrees to perform a second round of diagonal diffusion to enhance the encryption scheme's resistance to differential attacks, finally obtaining a first pure ciphertext matrix; performing 2D IWT on the RGB three channels of the color carrier image to obtain low-frequency and high-frequency sub-bands; using the least significant bit algorithm, converting the first pure ciphertext matrix into a bitstream and adaptively embedding it into the high-frequency sub-band, while performing 2K correction to compensate for amplitude errors; performing inverse 2D IWT on each modified frequency domain sub-band to generate and output a color encrypted image.

[0080] In this embodiment, as Figure 5 As shown, the input image is first pre-sorted and scrambled. Specifically, the original grayscale plaintext image "Mandril.tif" of size N×N is first read using the imread function, and its data type is converted to double to adapt to the subsequent calculation requirements; at the same time, the image of size N×N is read... The color carrier image "Lena_color.tif" is used as a backup. Next, the chaotic sequence S1 output by the chaotic sequence generation module is sorted in ascending order using the sort function to obtain its corresponding index mapping sequence (Sidx). After flattening the two-dimensional plaintext image into a one-dimensional row vector, the physical positions of the pixels are rearranged according to Sidx, thereby completely destroying the local spatial correlation of the original image. Then, the reshape function is used to restore it to a two-dimensional image to obtain the pre-set scrambled image P1.

[0081] Then, 2D CS measurement and quantization operations based on SVD optimization are performed on the pre-set scrambled image to obtain the dimensionality-reduced ciphertext data. Specifically, to improve the orthogonality and reconstruction accuracy of the measurement matrix, the chaotic sequence S2 is reshaped into measurement matrices Φ1 and Φ2 (of size M×N), and then singular value decomposition ([U,~,V]=svd(Φ), Φ=U*V, where U represents the left singular vector matrix of the measurement matrix and V represents the right singular vector matrix of the measurement matrix) is performed to extract the orthogonal basis mixture matrix as the optimized row measurement matrix Φ1 and column measurement matrix Φ2; subsequently, matrix multiplication ( , where T represents the transpose of the matrix) performs 2D CS on the preset scrambled image. This operation effectively avoids memory overflow caused by the traditional one-dimensional compressed sensing matrix flattening. After the measurement is completed, the round function is used to linearly scale the matrix in combination with the maximum and minimum values, and it is strictly quantized and mapped to the first compressed ciphertext matrix P2 of integers in the range of 0-255.

[0082] Next, variable-step and-direction DLJPS is performed on the quantized compressed matrix to achieve depth-based physical location obfuscation. Specifically, chaotic sequences S3 and S4 are used as control parameters, and floor and modulo operations are combined to dynamically generate the traversal step size of the Josephus outer loop (mod(floor( ),M)+1) and the starting direction (mod(floor( ),2)), and cross-reuse to generate the step size of the inner loop (mod(floor(S4*10)). 10 ),M)+1) and direction parameters (mod(floor(S4*10) 10 (2)); Based on the dynamic parameters described above, the double-layer Josephus ring logic is used to extract and rearrange each pixel of the first compressed ciphertext matrix P2 with varying step size and direction. This operation extends the traditional one-dimensional cyclic decoupling to a two-dimensional composite space, significantly increasing the complexity and unpredictability of scrambling, resulting in a scrambling matrix P3.

[0083] Subsequently, a bidirectional diagonal diffusion operation is performed to effectively resist differential attacks. Specifically, a mask matrix K of the same size as the scrambling matrix P3 is generated using the chaotic sequence S5 through the reshape function and modulo operation. Combining the current pixel and its adjacent ciphertext pixels to its left, top, and top left, a first round of diagonal modulo diffusion is performed from top left to bottom right (adding the current element with the corresponding elements to its left, top, top left, and K, and taking the modulo). Then, the matrix that has completed one round of diffusion is rotated 180 degrees using the rot90(..., 2) function, and a second round of diagonal modulo diffusion is performed from bottom right to top left. This bidirectional diagonal diffusion method breaks the limitations of unidirectional diffusion, allowing extremely small single-pixel changes in the plaintext to spread rapidly globally, achieving a perfect avalanche effect and generating the final first pure ciphertext matrix CI.

[0084] The method for single-pass diagonal diffusion is as follows:

[0085] ,

[0086] in, This represents an image before a single round of diffusion. Let represent the encrypted image obtained after one round of diffusion, K represent the diffusion mask matrix generated by the chaotic sequence, M represent the size of the image, and (m, n) represent the coordinates of the current pixel.

[0087] Finally, lossless steganography and 2K correction in the frequency domain are performed to generate a visually secure encrypted image. The RGB channels of the color carrier image are decomposed in the frequency domain using 2D IWT to extract the low-frequency approximate subband LL and the high-frequency detail subbands LH, HL, and HH containing the main energy. The first pure ciphertext matrix CI is converted into a binary data stream using operations such as dec2bin, and directly written into the LH, HL, and HH high-frequency subbands of the carrier in the form of least significant bits (LSB) according to a preset adaptive embedding depth (e.g., setting LH and HL to embed 1 bit, HH to embed 2 bits). Then, the correct2k correction algorithm is called to adaptively compensate for carry or borrow in the modified high-frequency subbands, eliminating the amplitude truncation error caused by LSB replacement. Finally, each frequency domain subband is reconstructed using the inverse integer wavelet transform (I2D IWT) function to generate and output a visually camouflaged color image X with normal physical meaning. This series of operations perfectly hides the ciphertext within a normal color image, fundamentally reducing the risk of malicious interception and targeted cryptanalysis of the system in public channels.

[0088] Step S4. Extract the ciphertext from the ciphertext image and perform reverse decryption and high-quality reconstruction based on the 2DPG-ED algorithm to restore the two-dimensional plaintext image.

[0089] The method for restoring a two-dimensional plaintext image includes: performing 2D IWT on the received ciphertext image, extracting and recombining the ciphertext bitstream from the high-frequency subband to obtain a second pure ciphertext matrix; based on the synchronously generated chaotic sequence, sequentially performing inverse diagonal diffusion and inverse DLJPS on the extracted second pure ciphertext matrix to recover the second compressed ciphertext matrix containing quantization errors; using the second compressed ciphertext matrix as the observation target, performing joint decompression and reconstruction using the 2DPG-ED algorithm, combining a total variation model in iterative gradient descent for sparse projection in the plaintext domain, filtering out reconstruction artifacts and channel noise, and outputting a high-quality decrypted image; generating an inverse index sequence based on the chaotic sequence, performing inverse sorting and scrambling restoration on the decrypted image, and outputting the final two-dimensional plaintext image.

[0090] In this embodiment, as Figure 6 As shown, the image decryption and joint reconstruction process will be detailed. The decryption process strictly follows the inverse symmetry of forward encryption and performs reverse parsing and optimized reconstruction of the data based on the mathematical principles of the specific algorithm. The specific decryption process is as follows:

[0091] (1) Frequency Domain Ciphertext Bitstream Extraction. This step is based on the lossless and reversible principle of 2D IWT (for example, in the forward transform, integers are mapped to integers using L=floor(A+B) / 2 and H=AB (where L represents the low-frequency sub-band, H represents the high-frequency sub-band, and A and B represent the pixel values ​​of two adjacent elements of the image, respectively), and the inverse transform can accurately restore the original values ​​through inverse algebraic operations) to extract the ciphertext. Specifically, the receiving end first uses I2D IWT to decompose the RGB three channels of the received color ciphertext image sequentially to obtain the corresponding low-frequency approximate sub-band and high-frequency detail sub-band (LH, HL, HH). According to the adaptive embedding depth parameter agreed upon during encryption, the steganographic binary bitstream is stripped. Subsequently, through binary-to-decimal numerical conversion, the discrete bitstream is re-aggregated and reshaped into an M×M second pure ciphertext matrix.

[0092] (2) Double reverse parsing of ciphertext space. This step is based on the structural symmetry of the encryption algorithm and the reflexivity of nonlinear mapping to counteract the effects of diffusion and scrambling in turn. The specific operations are as follows: ① Reverse bidirectional diagonal diffusion: The mask matrix K is recovered using the synchronously generated chaotic sequence S5. According to the structural symmetry of forward diagonal diffusion, the reverse operation is used during decryption: First, the rot90(..., 2) function is called to rotate the pure ciphertext matrix by 180 degrees. The second round of diffusion effects is counteracted by a simple reverse modular subtraction operation mod(ciphertext value-key value-adjacent ciphertext value, 256). After rotating 180 degrees again to restore the physical direction, the first round of reverse diffusion modular subtraction is performed according to the same rules to restore the intermediate reverse diffusion matrix. ② Reverse double-layer Josephus scrambling: The forward traversal step size and direction parameters are dynamically generated using the reconstructed chaotic sequence S3 and chaotic sequence S4 to deduce the global forward index of the two-dimensional physical position mapping. The reverse index (inv_idx) of the reverse mapping can be directly obtained by sorting the index. After flattening the intermediate inverse diffusion matrix, spatial shifting is performed using the inverse index to complete the reset. Finally, the matrix is ​​converted to double type using the linear extremum dequantization formula to recover the second compressed ciphertext matrix MI2.

[0093] (3) Joint decryption and reconstruction loop based on the 2DPG-ED algorithm. This step is based on the optimization principle of compressed sensing reconstruction and the sparse prior principle of natural images, embedding the decryption operator into the reconstruction iteration. A single iteration specifically includes the following sub-steps:

[0094] ① Encrypted Domain Gradient Calculation: Based on matrix differentiation and back projection principles. Orthogonal row and column measurement matrices Φ1 and Φ2 are constructed using the chaotic sequence S2 via singular value decomposition (SVD). The observation residuals of the k-th iteration are calculated. Using the transpose of the measurement matrix ( Using Φ2) as the back projection operator, the M×M residual is mapped back to the N×N ciphertext space through matrix multiplication, and gradient update is performed to obtain the intermediate matrix. .

[0095] ② Embedded Decryption: The inverse one-dimensional index sequence (inv_idx1) is obtained by using the sort function to solve for the chaotic sequence S1 corresponding to the pre-scrambled encryption. This is then directly applied to the noisy ciphertext matrix. Perform inverse sorting and scrambling, mapping it back to the plaintext space with local physical correlation, to obtain the plaintext structure estimation matrix. .

[0096] ③ Plaintext Domain Sparse Projection Denoising: Based on the gradient sparsity of natural plaintext images, a total variation (TV) regularization model is introduced in the plaintext domain. The plaintext structure estimation matrix is ​​obtained by solving for the minimum TV norm of the global image. Edge-preserving smoothing filtering is performed to remove reconstruction artifacts and channel noise during the iteration process, resulting in a smoothly updated image. .

[0097] ④ Forward scrambling loop closure: Update the denoised and smoothed image. The chaotic sequence S1 is sorted and scrambled again using its forward index, and then mapped back to the ciphertext field to generate the ciphertext. This is used to participate in the residual calculation of the next iteration.

[0098] (4) Convergence criterion and original plaintext recovery: Calculate the difference between the image matrices of two adjacent iterations (i.e., the Frobenius norm) in the loop. The iteration stops when the norm falls below the preset convergence threshold ε. Data range correction is then performed on the converged matrix, and it is converted to uint8 format to output the final two-dimensional plaintext image.

[0099] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. An image encryption and visual hiding method based on chaotic neural networks and compressed sensing, characterized in that, Includes the following steps: Construct a fractional-order heterogeneous chaotic neural network, and use the fractional-order chaotic neural network to put the encryption system into a complex chaotic state; Starting with an initial value, a chaotic sequence required for encryption is generated through numerical iteration; Based on the chaotic sequence, the plaintext image is subjected to two-dimensional compression and hybrid encryption, and then embedded into a color carrier to obtain a encrypted image; Ciphertext is extracted from the ciphertext image, and reverse decryption and high-quality reconstruction are performed based on the 2DPG-ED algorithm to restore the two-dimensional plaintext image; The method for constructing the fractional-order heterogeneous chaotic neural network includes: Define the fractional Caputo operator: ， Where q represents the fractional order. Let q represent the gamma function, and m represent the smallest integer greater than or equal to q. Describe the objective function The lower limit of time is The global operator of q-order Caputo fractional calculus at time t. Indicates the lower limit of time. Riemann-Liouville fractional integral operator of order mq This represents a standard m-order integer differential operator, where t represents the current evolution time. This represents the m-th derivative of the objective function with respect to the historical time variable τ. This represents the denominator of the power-law memory kernel function in fractional calculus formulas. This means that when the order q is exactly equal to an integer m, Caputo fractional calculus seamlessly degenerates into the objective function. The standard m-th order ordinary differential with respect to time t; A sinusoidal magnetically controlled memristor model is introduced to simulate the synaptic crossing connections of neurons in the brain region. The sinusoidal magnetically controlled memristor model is as follows: ， Where i represents the output current, v represents the input voltage, φ represents the internal magnetic flux state variable, α represents the activation gradient of the memristor, and k represents the internal control parameters of the memristor. This indicates that the state variable φ is from The initial q-order fractional derivative; By combining different nonlinear activation functions, and based on the fractional-order Caputo operator and the sinusoidal magnetically controlled memristor model, the fractional-order heterogeneous chaotic neural network is constructed: ， in, and This represents the neuron state variables using the Hardtanh activation function. and Let represent the neuron state variables using the LeakyReLU activation function, u represent the internal state variables of the memristor, p represent the coupling strength of the memristor, I represent the external input bias current, α represent the memristor activation gradient, and k represent the internal parameters of the memristor. This indicates that the state variable x starts from the initial time. The q-order Caputo fractional differential operator.

2. The image encryption and visual hiding method based on chaotic neural networks and compressed sensing according to claim 1, characterized in that, The Hardtanh activation function is: ； The LeakyReLU activation function is: ， Where β represents the leakage coefficient.

3. The image encryption and visual hiding method based on chaotic neural networks and compressed sensing according to claim 1, characterized in that, The method for generating the chaotic sequence includes: Starting with given initial values ​​and system parameters, the fractional-order heterogeneous chaotic neural network is discretized and iteratively solved. Discarding the transient processes of the first few iterations, extract the sequence of state variables of the system to obtain several sets of chaotic sequences of independent length.

4. The image encryption and visual hiding method based on chaotic neural networks and compressed sensing according to claim 1, characterized in that, The method for obtaining the dense image includes: The plaintext image is sorted by one-dimensional position index using the chaotic sequence to obtain a pre-set scrambled image; A measurement matrix is ​​constructed based on the chaotic sequence. The measurement matrix is ​​optimized by SVD singular value decomposition. Then, the preset scrambled image is subjected to 2D CS and linear quantization to obtain the first compressed ciphertext matrix. The step size and direction parameters are dynamically generated using the chaotic sequence, and a double-layer Josephus scrambling with variable step size and direction is performed on the first compressed ciphertext matrix. The scrambled matrix is ​​subjected to the first round of diagonal diffusion from the top left to the bottom right. Then the matrix is ​​rotated 180 degrees to perform the second round of diagonal diffusion to enhance the encryption scheme’s resistance to differential attacks, and finally the first pure ciphertext matrix is ​​obtained. Perform 2D IWT on the RGB three channels of the color carrier image to obtain the low-frequency subband and the high-frequency subband; The least significant bit algorithm is used to convert the first pure ciphertext matrix into a bit stream and adaptively embed it into the high-frequency subband, while 2K correction is performed to compensate for amplitude error. Perform inverse 2D IWT on each modified frequency domain subband to generate and output the color-coded dense image.

5. The image encryption and visual hiding method based on chaotic neural networks and compressed sensing according to claim 4, characterized in that, The method of diagonal diffusion is as follows: ， in, This represents an image before a round of diffusion. Let represent the encrypted image obtained after one round of diffusion, K represent the diffusion mask matrix generated by the chaotic sequence, M represent the size of the image, and (m, n) represent the coordinates of the current pixel.

6. The image encryption and visual hiding method based on chaotic neural networks and compressed sensing according to claim 4, characterized in that, The method for restoring the two-dimensional plaintext image includes: Perform 2D IWT on the received ciphertext image, extract the ciphertext bitstream from the high-frequency subband and reassemble it to obtain a second pure ciphertext matrix; Based on the synchronously generated chaotic sequence, reverse diagonal diffusion and reverse DLJPS are sequentially performed on the extracted second pure ciphertext matrix to recover the second compressed ciphertext matrix containing quantization error. Using the second compressed ciphertext matrix as the observation target, the 2DPG-ED algorithm is used to perform joint decompression and reconstruction. In the iterative gradient descent, the total variation model is combined to perform sparse projection in the plaintext domain, filter out reconstruction artifacts and channel noise, and output a high-quality decrypted image. A reverse index sequence is generated based on the chaotic sequence, and the decrypted image is reverse sorted and scrambled to restore it, outputting the final two-dimensional plaintext image.

Images